This group theory problem I found in an old math text. I am trying to
write a formal math proof to get practice writing proofs. Thanks.

Let G be a group. Let Aut(G) be a group of automorphisms. Let G1 and
G2 be groups and let f: G2 --> Aut(G1) be a homomorphism. The mapping
f defines a group structure on the set of ordered pairs G1 x G2. Let *
be a binary operation such that
       
       (g1, g2) * (h1, h2) = ( g1 [f(g2)](h1),  (g2)(h2) )     

Here, (g1, g2) and (h1, h2) belong to G1 x G2 and [f(g2)](h1) is an
element of G1. (So then the first coord. is a product of g1 and
[f(g2)](h1).)

Q1) If f is the identity map, find the formula for *
Q2) Prove that for any map f, the binary operation * gives a group
structure on G1 x G2.
Q3) a.) Show { (e1, g2) } is a subgroup of G1 x G2 under with the
operation *
        b.) Now show that { (g1,e2) } is a normal subgroup.
 
Q4) a.) Find all the homomorphisms from Z/3Z (the residue class mod 2)
to Aut(Z/7Z).      Find a noncommutative group of order 21 using one
of these homomorphisms.
       b.) Do the same for Z/2Z to Aut(Z/5Z) and find a noncommutative
group of order 10.

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Request for Question Clarification by mathtalk-ga on 19 Dec 2002 06:39 PST

Hi, hockeyfan213-ga:

The group construction you ask about is an (outer) semidirect product.
 The topic is a standard one:

http://www.wikipedia.org/wiki/Semidirect_product

There are some expert researchers here (myself included) who would be
interested in answering your question, but some clarification is in
order.

You state that you are trying "to get practice writing proofs".  I'm
certainly willing to help with that, but I'm not clear what sort of
response would be useful to you.

If you have attempted the parts of the question shown, and reached
some particular difficulty in one part or the other, it would perhaps
be expeditious for you to explain your work up to that point and why
you feel that an obstacle has been encountered.

Or perhaps you have not yet attempted these questions and would like
some "hints" about good "lines of attack".  Again I would be willing
to help.

On the other hand if you are seeking a very detailed proof of each
part of this exercise, I would ask you to review the Google Answers
pricing guidelines:

https://answers.google.com/answers/pricing.html  
 
Multiple part questions such as you ask here would normally have a
higher list price.  If you both raise your price and also post a
clarification here, the system will alert me and I can take another
look.

regards, mathtalk-ga

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Clarification of Question by hockeyfan213-ga on 19 Dec 2002 09:51 PST

Thanks for the tip about semi-direct products. It's very helpful.

I should have stated my question more explicitly: my question asks for
a formal proof as an answer. Sorry for the confusion.

(by the way, I did review google's pricing guidelines and I think some
experts would be able to crank out these proofs within half an hour;
as you said, it is a standard topic)

Theo.